Localization properties of quarks Philippe de Forcrand ETH Zrich - - PowerPoint PPT Presentation

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Localization properties of quarks Philippe de Forcrand ETH Zrich - - PowerPoint PPT Presentation

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SB Motivation Results Conclusion Localization properties of quarks Philippe de Forcrand ETH Zrich and CERN Outline 1. Motivation: QCD vacuum structure and SB 2. Puzzling results about fermion localization 3. A theoretical puzzle:

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university-logo Motivation

χSB

Results Conclusion

Localization properties of quarks

Philippe de Forcrand ETH Zürich and CERN

Outline

  • 1. Motivation: QCD vacuum structure and χSB
  • 2. Puzzling results about fermion localization
  • 3. A theoretical puzzle: the correlator of top. charge density
  • 4. Trying to put the pieces together

See hep-lat/0611034

GGI, Florence, June 2008

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Extra dimensions Vacuum structure

Motivation I: extra dimensions

  • Extra dimensions not seen ⇒ localization in 4d

Feasible by topological defect Rubakov & Shaposhnikov, 1983 fluctuations around classical “kink” solution are localized

→ lower-dimension effective field theory

Many more: Hosotani, Randall & Sundrum, Dvali & Shifman,....

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Extra dimensions Vacuum structure

Motivation I: extra dimensions

  • Extra dimensions not seen ⇒ localization in 4d

Feasible by topological defect Rubakov & Shaposhnikov, 1983 fluctuations around classical “kink” solution are localized

→ lower-dimension effective field theory

Many more: Hosotani, Randall & Sundrum, Dvali & Shifman,....

  • Localization at work:

Domain-Wall fermions in lattice QCD: 5d → 4d Kaplan 1992 Note: A5 = 0 frozen.

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Extra dimensions Vacuum structure

Motivation II: QCD vacuum structure

  • “Understand” confinement → identify relevant IR degrees of freedom
  • Confinement is non-perturbative → caused by topological excitations?

Candidates:

  • instantons

’t Hooft Codimension 4: point-like topological obstruction

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Extra dimensions Vacuum structure

Motivation II: QCD vacuum structure

  • “Understand” confinement → identify relevant IR degrees of freedom
  • Confinement is non-perturbative → caused by topological excitations?

Candidates:

  • Abelian monopoles

’t Hooft Aµ → adjoint Higgs → BPS monopole Codimension 3: line-like topological obstruction

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Extra dimensions Vacuum structure

Motivation II: QCD vacuum structure

  • “Understand” confinement → identify relevant IR degrees of freedom
  • Confinement is non-perturbative → caused by topological excitations?

Candidates:

  • center vortices

Mack, ’t Hooft Codimension 2: ZN singular transformation on sheet

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Extra dimensions Vacuum structure

Motivation II: QCD vacuum structure

  • “Understand” confinement → identify relevant IR degrees of freedom
  • Confinement is non-perturbative → caused by topological excitations?

Candidates: instantons, Abelian monopoles, center vortices

  • All objects are “thick”: size O (1/ΛQCD)
  • Should also explain chiral symmetry breaking/restoration

Identify correct candidate by lattice measurements In the past: need to filter out UV fluctuations to see structure Smoothing/cooling/smearing to reduce action Evolve towards action minimum, ie. classical solution → instantons Can one avoid such bias?

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Chiral symmetry breaking/restoration

Anderson 1958: random tight-binding Hamiltonian Random impurities, each with -localized bound e−

  • random interaction energy with crystal ions

How does conductivity depend on overlap of bound states ? Eigenstates of H = ∆+ν

∆: discretized (lattice) Laplacian (hopping); ν: random potential

Localization ≡ eigenmode |ψ(r)|2 ∼ exp(−r) for r → ∞ with prob. 1

→ no electric conductivity

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Anderson transition: H = ∆+ν

  • Result: localization if - disorder sufficiently large or
  • energy sufficiently low

E very large → plane waves E very small → hopping to all neighbouring sites forbidden

  • Spectrum:

mobility edge

λ λ c

localized extended bulk

E <

λc →

localized > extended pFermi <

λc →

insulator > conductor Transition driven by temperature, or by disorder (T = 0, quantum)

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Anderson variations

  • 1. Low dimension:

d = 1: all states localized for any disorder d = 2: same Lee & Ramakrishnan, RMP 1985

  • 2. Modify Hamiltonian: H = ∆+ν:
  • randomness in hopping term ∆:

qualitatively similar

  • make ∆ long-range:

∆ij ∝

1

|rij|α long-range

Result: transition for α = d (α > d

→

localization) Mirlin 1996 d = 3 ←

→ dipole-dipole interactions

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Chiral symmetry breaking à la Diakonov & Petrov (1984)

  • Recall Banks-Casher: ¯

ψψ = limm→0 limV→∞ −π ρ(0)

How to obtain density of zero-modes ?

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Chiral symmetry breaking à la Diakonov & Petrov (1984)

  • Instanton supports chiral Dirac zero-mode

’t Hooft Superposition of I’s and A’s? D

/(AI

µ)ψI = 0

but D

/(∑I,A AI,A

µ )ψI = 0

zero-modes → displaced

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Chiral symmetry breaking à la Diakonov & Petrov (1984)

  • Instanton supports chiral Dirac zero-mode

’t Hooft Superposition of I’s and A’s? D

/(AI

µ)ψI = 0

but D

/(∑I,A AI,A

µ )ψI = 0

zero-modes → displaced

  • New eigenmodes?

Write Dirac operator in basis of original I,A zero-modes ψI,ψA: D

/ =

  • TIA

T †

IA

  • zero-diagonal because of chirality

Overlap Tij = ψI

i|ψA j ∼ 1

|rij|3 in d = 4 → delocalization

Support of eigenmodes ∼ I,A

  • Eigenvalues ∼ uniformly spread in [−ˆ

λ,+ˆ λ], ˆ λ ≈ ¯

rI

¯

RIA

χSB: ¯ ψψ ∼ −1

¯

rI ¯ R2

IA

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Chiral symmetry breaking à la Diakonov & Petrov (1984)

  • Instanton supports chiral Dirac zero-mode

’t Hooft Superposition of I’s and A’s? D

/(AI

µ)ψI = 0

but D

/(∑I,A AI,A

µ )ψI = 0

zero-modes → displaced

  • New eigenmodes?

Write Dirac operator in basis of original I,A zero-modes ψI,ψA: D

/ =

  • TIA

T †

IA

  • zero-diagonal because of chirality

Overlap Tij = ψI

i|ψA j ∼ 1

|rij|3 in d = 4 → delocalization

Support of eigenmodes ∼ I,A

  • Eigenvalues ∼ uniformly spread in [−ˆ

λ,+ˆ λ], ˆ λ ≈ ¯

rI

¯

RIA

χSB: ¯ ψψ ∼ −1

¯

rI ¯ R2

IA

  • Phenomenology reproduced with ¯

r ∼ 0.3 fm,

¯

RIA ∼ 1 fm Instanton liquid Shuryak, Schaefer

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Dirac eigenmodes on the lattice

hep-lat/9810033 PdF et al. lowest eigenmode of staggered D

/

no cooling Eigenmode support ∼ Instanton + Antiinstanton

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Comparison with Anderson

  • Difference:
  • Dirac eigenvalues come in pairs ±iλ

(plus zero)

  • interested in spectral properties (eg. eigenvalue repulsion) around 0
  • ie. middle of spectrum ↔

edge of spectrum for Anderson (bosons)

  • modeled by chiral random matrix ensemble

Garcia-Garcia

  • Similarity:

possible “depercolation” transition to localized states → ρ(0) = 0 Then ¯

ψψ = 0: chiral symmetry restored from small changes in TIA

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Chiral symmetry restoration at finite temperature

  • Shuryak: detD

/ → time-oriented I − A molecules

  • transition in quenched theory?
  • I − A molecules not seen on lattice
  • Diakonov & Petrov: more subtle
  • g(T) ց ⇒ instanton action ր ⇒ density of I,A decreases
  • TIA ∼ exp(−πRIAT)

decreased overlap → transition to localization

1/T

T = 0 T > 0

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Vacuum structure from eigenmode |ψ(x)|2?

  • Diakonov-Petrov χSB scenario does not require instantons
  • nly chiral zero-modes
  • Compatible with other topological defects:

domain-walls, monopoles, vortices,.. Reinhardt [chiral zero-mode on any topological defect? Q non-integer]

  • Working assumption:

extended modes have support on topological defects

⇓

deduce vacuum structure from spatial distribution of eigenmode gauge invariant; no smoothing/cooling/smearing...

  • Dirac fermions
  • can compare with bosons in various representations
  • Surprises; work in progress
  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Main tool: Inverse Participation Ratio

  • Definition:

IPR ≡ V

∑x |ψ(x)|4 (∑x |ψ(x)|2)2

(ratio of moments)

  • Simple cases:
  • |ψ(x)| = 1 ∀x

= ⇒

IPR = 1

  • |ψ(x)| = δx,x0

= ⇒

IPR = V

  • |ψ(x)| = 1
  • n fraction f of sites

= ⇒

IPR = 1

f

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

IPR: what to expect?

IPR ∼ 1/fraction of occupied lattice sites

  • any “thick” (macroscopic) object:

IPR = constant

  • “thin” (singular) instantons, monopoles, vortices:

IPR → ∞ as a → 0

  • ccupied sites ∝ a−1

total sites ∝ a−2

  • ccupied fraction f = a+1

IPR ∼ 1

f ∝ a−1

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion Anderson Diakonov-Petrov IPR

Strategy

  • Case of interest:

|ψ(x)| = 1 on manifold of dim. d, “volume” V d

Fraction of lattice sites f = V d/ad

V/a4

(a lattice spacing, V 4-volume) IPR ∝ ad−4

→

determine d by scaling of IPR versus a d = 0,1,2 =

⇒

“thin” instantons, monopoles, vortices

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

IPR measurement I

  • SU(3), quenched, Symanzik gauge, Asqtad D

/ (no exact zero-modes)

  • IPR → constant as V → ∞ (?)
  • a dependence: IPR diverges as a → 0

(Note scale of IPR)

2 2.5 3 3.5 4 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 <IPR> Lattice spacing (Fermi) <IPR> 0.28/a^(0.93) + 0.95 c/a + c

d = 3.07± 0.15 Hetrick et al. (MILC) hep-lat/0410024 + 0510025 d = 3 → eigenmodes localized on domain-walls of thickness ≪ 0.1 fm branes on the lattice!

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

IPR measurement II

  • SU(2), quenched, Wilson gauge, overlap D

/ (→ exact zero-modes)

  • IPR = b0 + b1V, ie. eigenmodes are localized on finite nb. of sites
  • IPR = c0 + c1 ad−4 with d = 0, ie. eigenmodes support is point-like

2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 18 IPR V, fm4 I0 Il 5 10 15 20 25 30 0.08 0.09 0.1 0.11 0.12 0.13 0.14 IPR a, fm I0 Il 0-modes near-0 modes 0-modes near-0 modes d = 0 1 2 3

Polikarpov, Zakharov et al., hep-lat/0505016 + 0510098 Note scale of IPR ≫ SU(3)

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Speculative interpretation (Zakharov)

  • ¯

ψψ and eigenvalues behave ’normally’ as a → 0 while IPR → ∞

evidence of fine-tuning energy vs entropy

  • confinement is caused by d = 2 “thin” center vortex sheets
  • topological density at point-like sheet intersections: εijklFijFkl
  • Circumstancial evidence:

Removing center vortices destroys confinement and χSB

5 10 15 20 25 50 100 150 200 250 300 350 400 Iλ λ, MeV Full VR

← IPR = 1

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

My conservative interpretation

  • d = 3 (SU(3)) vs d = 0 (SU(2)): lack of universality at short distance?

Defects at scale a may become dense depending on details of lattice action dislocations (size a instantons) in SU(2) Pugh & Teper ’89 Energy vs entropy:

  • energetic suppression: exp(− 4

g2

0 S∗)

  • entropic enhancement: nb. of positions Vphys./a4 ∼ exp(+ β1

2β0 1 g2

0 )

Result:

  • Entropy wins for SU(2) with Wilson action
  • Energy wins (dislocations suppressed) for SU(3) with Symanzik action

Problem cured by adding irrelevant terms in lattice action

⇒ forget d = 0 result. Can one understand d = 3 “branes”?

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Why d = 4 ?

d = 4 is the dimension of macroscopic, classical objects, BUT:

  • A kink does not look smooth as a → 0
  • Ph. de Forcrand

GGI, June 2008 Localization

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SLIDE 27

university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Why d = 4 ?

d = 4 is the dimension of macroscopic, classical objects, BUT:

  • A kink does not look smooth as a → 0
  • Ph. de Forcrand

GGI, June 2008 Localization

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SLIDE 28

university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Why d = 4 ?

d = 4 is the dimension of macroscopic, classical objects, BUT:

  • A kink does not look smooth as a → 0
  • Interface in 3d Ising model:

genus diverges as a → 0 Caselle, Gliozzi, Vinti ’93 Quantum fields are rough → d < 4. Why d = 3 ?

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Correlator of topological charge density

  • Continuation Minkowski ↔ Euclidean ⇒ q(0)q(x = 0)Eucl. < 0

(reflection positivity, or q ∼ E · B → i E · B) Seiler & Stamatescu

  • But
  • d4x q(0)q(x) = χtop ∼ (190MeV)4 ⇒

contact term

  • q(x) has canonical dim. 4 →
  • d4x 1/|x|8

UV-divergent Divergence cancelled by contact term → “fine tuning”

|x|

  • 1/|x|8

delta(0)

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Correlator of topological charge density

  • Continuation Minkowski ↔ Euclidean ⇒ q(0)q(x = 0)Eucl. < 0

(reflection positivity, or q ∼ E · B → i E · B) Seiler & Stamatescu

  • But
  • d4x q(0)q(x) = χtop ∼ (190MeV)4 ⇒

contact term

  • q(x) has canonical dim. 4 →
  • d4x 1/|x|8

UV-divergent Divergence cancelled by contact term → “fine tuning”

|x|

  • 1/|x|8

delta(0)

0.2 0.4 0.6 0.8 rfm 1.101 2.101 3.101 4.101 1.101 2.101 3.101 q0qrfm8 Topological charge density correlator a0.1650 fm a0.1100 fm a0.0825 fm

q(x) = Trcolor,Diracγ5D

/ Horvath et al., hep-lat/0504005 Also Schierholz et al., hep-lat/0509164

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Antiferromagnetic structure for q(x) ?

  • Kentucky group (Horvath et al.):

sign(q(x)) forms 2 space-filling 3d structures (transverse size O (a)) Koma, Ilgenfritz, Schierholz et al., hep-lat/0509164

  • Reproduced by effective anti-ferromag. model Boyko & Gubarev

Same for 1 1 d CP3 model Thacker et al., hep-lat/0509066

  • Ph. de Forcrand

GGI, June 2008 Localization

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SLIDE 32

university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Antiferromagnetic structure for q(x) ?

  • Kentucky group (Horvath et al.):

sign(q(x)) forms 2 space-filling 3d structures (transverse size O (a)) Koma, Ilgenfritz, Schierholz et al., hep-lat/0509164

  • Reproduced by effective anti-ferromag. model Boyko & Gubarev

Same for 1 1 d CP3 model Thacker et al., hep-lat/0509066

  • Ph. de Forcrand

GGI, June 2008 Localization

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SLIDE 33

university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Antiferromagnetic structure for q(x) ?

  • Kentucky group (Horvath et al.):

sign(q(x)) forms 2 space-filling 3d structures (transverse size O (a)) Koma, Ilgenfritz, Schierholz et al., hep-lat/0509164

  • Reproduced by effective anti-ferromag. model Boyko & Gubarev
  • Same for (1+ 1)d

CP3 model Thacker et al., hep-lat/0509066

  • Ph. de Forcrand

GGI, June 2008 Localization

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SLIDE 34

university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Antiferromagnetic structure for q(x) ?

  • Kentucky group (Horvath et al.):

sign(q(x)) forms 2 space-filling 3d structures (transverse size O (a)) Koma, Ilgenfritz, Schierholz et al., hep-lat/0509164

  • Reproduced by effective anti-ferromag. model Boyko & Gubarev
  • Same for (1+ 1)d

CP3 model Thacker et al., hep-lat/0509066

  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Trying to make sense of it all...

  • To the rescue: Koma, Ilgenfritz, Schierholz et al., hep-lat/0509164

SU(3), Lüscher-Weisz gauge (no disloc.), overlap D

/ (exact zero-modes)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 10 15 20 25 30 IPR Lphys/a MILC Koma

  • IPR numerically similar to MILC (for non-zero modes)
  • scaling consistent with d = 3

Unifying interpretation:

  • evidence for localization of quarks on d = 3 domain-walls:

consistent with observed topological charge domains

  • update: multi-fractal

Ilgenfritz et al., 0705.0018

  • Ph. de Forcrand

GGI, June 2008 Localization

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SLIDE 36

university-logo Motivation

χSB

Results Conclusion IPR

  • Top. correlator

Synthesis

Spatial correlator of Dirac eigenmode

  • Check spatial structure of 3d support

IF eigenmode |ψ(x)| = 1 on 3d fractal, 0 elsewhere then |ψ(0)||ψ(x)| ∼ 1/|x|

1 2 0.16 0.4 1 2.5 <p(x)p(x+r)> r (Fermi) a=0.163 a=0.128 a=0.110 a=0.0915 1/|x|

Hetrick et al. (MILC), hep-lat/0510025

  • not inconsistent?
  • fractal structure stops at |x| ∼ 1/ΛQCD
  • Ph. de Forcrand

GGI, June 2008 Localization

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university-logo Motivation

χSB

Results Conclusion

Conclusions

  • Quantum fields not smooth: classical lumps ↔ quantum descendents
  • Wild goose chase? learn nothing about “structure” [at scale 1/ΛQCD]

by looking at UV distances

  • UV? Theoretical argument + some numerical evidence → sandwich

alternating 3d layers of diverging ± topological charge density Bizarre but allowed?

  • not inconsistent with instanton, monopole, vortices at scale 1/ΛQCD

Vacuum structure depends on scale

  • Ph. de Forcrand

GGI, June 2008 Localization